1ok...what about the first one?
1.\int \frac{dx}{tanx+secx+sinx+cosx+cotx+cosecx}
2.U_{n}=\int_{0}^{1}{x^{n}\left(2-x \right)^{n}dx}, V_{n}=\int_{0}^{1}{x^{n}\left(1-x \right)^{n}dx} ; prove\ \right| that\ \right| U_{n}=2^{2n}V_{n}
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4 Answers
3412) U_n = \int_0^1 x^n (2-x)^n \ dx \Rightarrow \frac{U_n}{2^{2n}} = \int_0^1 \left(\frac{x}{2}\right) ^n \left(1-\frac{x}{2} \right)^n \ dx
= 2\int_0^ {\frac{1}{2}} t ^n \left(1-t \right)^n \ dt ; t = \frac{x}{2}
= \int_0^ {\frac{1}{2}} t ^n \left(1-t \right)^n \ dt +\int_0^ {\frac{1}{2}} t ^n \left(1-t \right)^n \ dt
The second integrand by the transformation t \rightarrow 1-t can be converted to
\int_{\frac{1}{2}}^ 1 t ^n \left(1-t \right)^n \ dt
Thus
\frac{U_{2n}}{2^{2n}} = \int_0^{\frac{1}{2}} t ^n \left(1-t \right)^n \ dt + \int_{\frac{1}{2}}^ 1 t ^n \left(1-t \right)^n \ dt = \int_0^1 t ^n \left(1-t \right)^n \ dt = V_n
and we are done
341http://targetiit.com/iit-jee-forum/posts/prob-17742.html
see if this helps.